They say the two men were involved in an apparent altercation which took place on the cycle path next to the River Nith at around 9.40 am on Thursday May 21.
Officers say the area was busy at the time and that one of the cyclists had a dog.
They ask that anyone with information contacts police on telephone number 101 or Crimestoppers anonymously on 0800 555 111.
NEWSDESK: 01461-202417
